AL Combined Maths Syllabus

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GENERAL CERTIFICATE OF EDUCATION
ADVANCED LEVEL
(Grade 12 and 13)
COMBINED MATHEMATICS
SYLLABUS
(Effective from 2017)
Department of Mathematics
Faculty of Science and Technology
National Institute of Education
Maharagama
SRI LANKA

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CombinedMathematics
Grade 12 and 13 - syllabus
NationalInstituteofEducation
First print 2017
ISBN :
DepartmentofMathematics
FacultyofScienceandTechnology
NationalInstituteofEducation
www.nie.lk
Printed by:

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CONTENTS
1.0 Introduction ......................................... iv
2.0 Common National Goals ......................................... v
3.0 Basic Competencies ..................................... vi
4.0 Aims of the syllabus ......................................... viii
5.0 Relationship between the common national
goals and the objectives of the syllabus ......................................... ix
6.0 Proposed plan in the allocation of competency
levels in the syllabus into the school terms ......................................... x
7.0 Proposed term wise breakdown of the syllabus ......................................... 1
8.0 Syllabus. ......................................... 5
9.0 Teaching learning strategies ......................................... 65
10.0 Assessment and Evaluation ......................................... 51
Mathematical symbols and notations ......................................... 65-68

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1.0 INTRODUCTION
The aim of education is to turn out creative children who would suit the modern world. To achieve this, the school curriculum should be revised according to
the needs of the time.
Thus, it had been decided to introduce a competency based syllabus in 2009. The earlier revision of the G.C.E. (Advanced Level) Combined Mathematics
syllabus was conducted in 1998. One of the main reason for the need to revise the earlier syllabus had been that in the Learning - Teaching- Assessment
process, competencies and competency levels had not been introduced adequately. It has been planned to change the existing syllabus that had been
designed on a content based approach to a competency based curriculum in 2009. In 2007, the new curriculum revision which started at Grades 6 and 10 had
introduced a competency based syllabi to Mathematics. This was continued at Grades 7 and 11 in 2008 and it continued to Grades 8 and 12 in 2009.
Therefore, a need was arisen to provide a competency based syllabus for Combined Mathematics at G.C.E.(Advanced Level) syllabus the year 2009.
After implementing the Combined Mathematics syllabus in 2009 it was revisited in the year 2012. In the following years teachers view’s and experts opinion
about the syllabus, was obtained and formed a subject comittee for the revision of the Combined Mathematics syllabus by acommodating above opinions the
committee made the necessary changes and revised the syllabus to implement in the year 2017.
The student who has learnt Mathematics at Grades 6-11 under the new curriculum reforms through a competency based approach, enters grade 12 to learn
Combined Mathematics at Grades 12 and 13 should be provided with abilities, skills and practical experiences for his future needs. and these have been
identified and the new syllabus has been formulated accordingly. It is expected that all these competencies would be achieved by pupils who complete
learning this subject at the end of Grade 13.
Pupils should achieve the competencies through competency levels and these are mentioned under each learning outcomes
It also specifies the content that is needed for the pupils to achieve these competency levels. The number of periods that are needed to implement the
process of Learning-Teaching and Assessment also mentioned in the syllabus.
Other than the facts mentioned regarding the introduction of the new curriculum, what had already been presented regarding the introduction of Combined
Mathematics Syllabus earlier which are mentioned below too are valid.

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• To decrease the gap between G.C.E. (Ordinary Level) Mathematics and G.C.E. (Advanced Level) Combined Mathematics.
• To provide Mathematics knowledge to follow Engineering and Physical Science courses.
• To provide a knowledge in Mathematics to follow Technological and other course at Tertiary level.
• To provide Mathematics knowledge for commercial and other middle level employment.
• To provide guidance to achieve various competencies on par with their mental activities and to show how they could be developed throughout
life.

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2.0 Common National Goals
The national system of education should assist individuals and groups to achieve major national goals that are relevant to the individual and society.
Over the years major education reports and documents in Sri Lanka have set goals that sought to meet individual and national needs. In the light of the
weaknesses manifest in contemporary educational structures and processes, the National Education Commission has identified the following set of goals
to be achieved through education within the conceptual framework of sustainable human development.
I Nation building and the establishment of a Sri Lankan identity through the promotion of national cohesion, national integrity, national unity, harmony
and peace, and recognizing cultural diversity in Sri Lanka’s plural society within a concept of respect for human dignity.
II Recognizing and conserving the best elements of the nation’s heritage while responding to the challenges of a changing world.
III Creating and supporting an environment imbued with the norms of social justice and a democratic way of life that promotes respect for human rights,
awareness of duties and obligations, and a deep and abiding concern for one another.
IV Promoting the mental and physical well-being of individuals and a sustainable life style based on respect for human values.
V Developing creativity, initiative, critical thinking, responsibility, accountability and other positive elements of a well-integrated and balance personality.
VI Human resource development by educating for productive work that enhances the quality of life of the individual and the nation and contributes to
the economic development of Sri Lanka.
VII Preparing individuals to adapt to and manage change, and to develop capacity to cope with complex and unforeseen situations in a rapidly changing
world.
VIII Fostering attitudes and skills that will contribute to securing an honourable place in the international community, based on justice, equality and mutual
respect.

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ii
3.0 Basic Competencies
ThefollowingBasicCompetenciesdevelopedthrough educationwill contributeto achievingtheaboveNational Goals.
(i) Competencies in Communication
Competencies in Communication arebased on foursubjects: Literacy, Numeracy, Graphicsand ITproficiency.
Literacy: Listenattentively, speckclearly, readformeaning, writeaccuratelyand lucidlyand communicateideas effectively.
Numeracy: Usenumbersforthings, spaceand time, count, calculateand measuresystematically.
Graphics: Make sense ofline and form, express and record details, instructions and ideas with lineform and color.
ITproficiency: Computeracyand theuseofinformationandcommunication technologies (ICT)inlearning,intheworkenvironmentandin
personallife.
(ii) Competencies relating to Personality Development
- Generalskillssuchascreativity,divergentthinking,initiative,decisionmaking,problemsolving,criticalandanalyticalthinking,team
work,inter-personalrelations,discoveringandexploring;
- Valuessuch asintegrity, toleranceand respectforhuman dignity;
- Emotionalintelligence.
(iii) Competencies relating to the Environment
Thesecompetencies relatetotheenvironment : social,biological and physical.
Social Environment : Awareness of the national heritage, sensitivityand skills linked to being members of a plural society, concern for
distributivejustice,socialrelationships,personalconduct,generalandlegalconventions,rights, responsibilities,duties andobligations.
Biological Environment :Awareness, sensitivity and skills linked to the living world, people and the ecosystem, the trees, forests, seas,
water,airandlife-plant,animalandhumanlife.

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PhysicalEnvironment:Awareness,sensitivityandskillslinkedtospace,energy,fuels,matter,materialsandtheirlinkswithhumanliving,food,
clothing, shelter, health,comfort, respiration, sleep, relaxation, rest,wastes and excretion.
Includedhereareskillsin usingtoolsandtechnologiesforlearning,workingandliving.
(iv) Competencies relating to Preparation for theWorld of Work.
Employmentrelatedskillstomaximizetheirpotential andtoenhancetheircapacity
to contributeto economicdevelopment,
to discover theirvocational interests ad aptitudes,
to choosea job that suits their abilities, and
to engage in arewardingand sustainablelivelihood.
(v) Competencies relating to Religion and Ethics
Assimilatingandinternalizingvalues,sothatindividualsmayfunctioninamannerconsistentwiththeethical,moralandreligiousmodesof
conductin everydayliving,selectingthatwhichismostappropriate.
(vi) Competencies in Play and the Use of Leisure
Pleasure, joy, emotions and such human experiences as expressed through aesthetics, literature, play, sports and athletics, leisure pursuits
and othercreativemodes ofliving.
(vii) Competencies relating to ‘ learning to learn’
Empoweringindividualstolearnindependentlyandtobesensitiveandsuccessfulinrespondingtoandmanagingchangethroughatransformative
process,in a rapidlychanging, complex and interdependent world.

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4.0 AIMS OFTHE SYLLABUS
(i) Toprovidebasicskills ofmathematicstocontinuehigherstudiesinmathematics.
(ii) To providethestudents experienceon strategies ofsolvingmathematicalproblems.
(iii) Toimprovethestudentsknowledgeoflogicalthinkinginmathematics.
(iv) To motivatethestudents to learn mathematics.
This syllabus was prepared to achieve theabove objectives through learning mathematics. It is expected not onlyto improve theknowledge of math-
ematics but also to improve the skill of applying the knowledge of mathematics in their day to daylife and character development through this new
syllabus.
When weimplement this competencyBasedSyllabus in thelearning-teachingprocess.
• Meaningful Discoverysituationsprovided would lead tolearningthat would bemorestudent centred.
• It will providecompetencies accordingto the level ofthestudents.
• Teacher's targets will be more specific.
• Teacher can provide necessaryfeed back as he/she is ableto identifythe student's levels of achieving each competencylevel.
• Teachercan playatransformation role bybeingawayfrom othertraditional teaching methods.
When thissyllabus is implemented intheclassroom theteachershould beabletocreatenewteachingtechniquesbyrelatingto varioussituations under
given topics accordingto the current needs.
Forthe teachersit would be easyto assess and evaluatethe achievement levels of students as it will facilitateto do activities on each competencylevel
in thelearning-teachingprocess.
In this syllabus,thesections given belowarehelpful in theteaching-learningprocessofCombined Mathematics.
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A Basic Course for G.C.E (Advanced Level)
Combind Mathematics
04
02
04
04
02
02
1. Review of
Basic
Algebra
1.1 Expands algebric
expressions
1.2 Factorises algebraic expres
sions
1.3 Simplifiesalgebraic
fractions
1.4 Solves Equations
1.5 Simplifiesexpressions
involving indices and
logarithims
1.6 Describes and uses the
properties of proportions
• Expansion of 2 2 3 3
,
a b a b
 
and 2 3
( ) , ( )
a b c a b c
   
• Factorisation for 2 2 3 3
,
a b a b
 
• Addition,Subtraction,Multiplication
andDivisionofAlgebraicfractions.
• Equationswithalgebraicfractions,
simultaneous equations up to three
unknowns , quadratic simultianeous
equations withtwo variabess.
• Rulesofindiciesfundamental
propertiesoflogarithies
• Equalityof tworatios is a proportion
: :
a c
a b c d
b d
  
Properties of the abve proporture
Appliestheformulatosimplifyalgebraic
expression.
• Factorises algebraicexpression by
usingtheformule.
• Uses the knowledgeof factorisation in
theformulaesinvolvedexpansion
• Solves equationsbyusingfactorisation
formulaesinvolvedexpansion
• Simplifiesexpressionsinvolvesindices.
• Solveequationswithindices.
• Simplifieslogarithimsexpressions.
• Solvesequationswithlogarathims
• Finds valuesofalgebraiacexpression
usingpropotions
• Solves equationsusingpropotions
Competency Competency Level Content Learning outcome
No. of
Periods

2
Competency Competency Level Contents Learning outcomes
No. of
Periods
2.1 Identifiestheoremsinvolving
rectangles in circle and uses
is Geometry problems.
2.2 Applies pythagoras theorem
and its extensions in prob
lems.
2.3 Applies bisector theorem in
geometry problems.
2.4 Applies theorems on
similartriangles in geometry.
2.5 Identifies the centres of a
triangles
• Pyhtagarustheoremaculeangled
theorem obtuseangled theorem
applloniuistheorem.
• Pythagoras theorem acuteangled
theorem obtuseangled theorem
Apolliniustheorem.
• Internal and externalAnglesofa
triangle bisectsthe oppositeside
proportially,
• Theareasofsimilartrianglesare
propotional to thesquare of the
correspondingsides.
• Circum centre, Incentre,
Orthocentre, Centroial medians,
attitudes.
• Describes thetheorem when two
chords intersect. and the theorem
involvedalternatesegments.
• Uses the above theorems to solve
problems.
• Uses the theorems to prove state-
ments.
• Uses thetheoremto find length and
angless.
• Usesthetheoreminvolvedfindlength
intriangle.
• Describes the theorem and uses it to
solveproblems.
• Definies the4centres ofatriangles
and usesit in problems.
04
04
02
03
02
2. Geometry
Analyses
plane geom-
etry
For Comments

6.0 PROPOSED TERM WISE BREAKDOWN OF THE SYLLABUS
Competency Levels Subject Topics Number of
Periods
Combined Mathematics I
1.1, 1.2 Realnumbers 02
2.1, 2.2 Functions 04
8.1, 8.2 Angularmeasurements 02
17.1, 17.2 Rectangularcartesiansystem, Straightline 03
9.1, 9.2, 9.3, 9.4 Circularfunctions 12
11.1 sinerule,cosinerule 01
4.1, 4.2, 4.3 Polynomials 07
10.1, 10.2, 10.3, 10.4 Trigonometricidentities 14
5.1 Rationalfunctions 06
6.1 Index lawsandlogarithmiclaws 01
7.1, 7.2, 7.3 Basicpropertiesof inequalitiesandsolutionsofinequalities 14
9.5 Solvingtrigonometricequations 04
Combined Mathematics II
1.1> 1.2> 1.3> 1.4 Vectors 14
2.1> 2.2> 2.3 Systems ofcoplanar forces acting at a point 10
3.1, 3.2 Quadraticfunctionsand quadraticequations 25
12.1, 12.2, 12.3 Inversetrigonometricfunctions 08
11.2 sinerule,cosinerule 06
First Term
Grade 12
Second Term
3
For Comments

18
23
23
30
15
08
08
10
10
14
15
Limits
System ofcoplanar forces acting on a rigid body
Motioninastraightline
Derivatives
Applicationsof derivatives
Projectiles
Equilibiriumof threecoplanarforces
Friction
Newton’s lawsofmotion
Work, power,energy,
Impulseandcollision
13.1, 13.2, 13.3, 13.4, 13.5, 13.6,
13.7, 13.8
2.4, 2.5, 2.6, 2.7
3.1, 3.2, 3.3
14.1, 14.2, 14.3, 14.4, 14.5, 14.6,
14.7, 14.8
15.1, 15.2, 15.3, 15.4
3.4
2.8
2.9
3.5
3.6,3.7
3.8,3.9
Periods
Third Term
4
For Comments

16
28
10
10
30
16
15
15
05
18
18.1, 18.2, 18.3, 18.4, 18.5
16.1, 16.2, 16.3, 16.4, 16.5,
16.6, 16.7, 16.8, 16.9
2.10
2.11
3.10,3.11,3.12,3.13
3.14, 3.15, 3.16
Combined Maths I
26.1, 27.1, 27.2, 27.3, 27.4,
27.5
24.1, 24.2, 24.3, 24.5
19.1
20.1, 20.2, 21.1, 21.2
Straightline
Intergration
Jointed rods
Frame works
Relativemotion
Circularrotion
Circle
PermutationsandCombinations
Principleof MathematicalInduction
Series
Grade 13
Periods
First Term
Second Term
5
For Comments

10
18
20
12
18
14
18
18
Probability
Simpleharmonicmotion
Center of mass
Binomialexpansion
Complexnumbers
Matrices
Probability
Statistics
4.1, 4.2
3.17, 3.18, 3.19
2.12, 2.13, 2.14, 2.15, 2.16, 2.17
22.1, 22.2, 22.3
23.1, 23.2, 23.3, 23.4, 23.5, 23.6
25.1, 25.2, 25.3, 25.4
4.3, 4.4, 4.5
5.1, 5.2, 5.3, 5.4, 5.5, 5.6
5.7, 5.8, 5.9.
Periods
Third Term
6
For Comments

7
For Comments
T
Combined Mathematics I 70
Combined Mathematics II 24 94
Subject Number of Periods Total
Second Term
First Term
Third Term
Fourth Term
Fifth Term
Sixth Term

8
No. of
Periods
7.0 Detailed Syllabus - COMBINED MATHEMATICS - I
01
01
02
1. Analyses
the system
of real
numbers
2. Analyses
single
variable
functions
1.1 Classifies theset of real
numbers
1.2 Uses surds or decimals to
describe real numbers
2.1 Review of functions
• Historicalevolutionofthenumber
system
• Notations for sets of numbers
'
, , , , ,
 
• Geometrical representation of real
numbers
Ú Numberline.
• Decimal representationofareal
number
Ú Decimals,infinitedecimals,
recurringdecimals,and
non-recurrringdecimals
• Simplificationof expressions
involvingsurds
• Intuitiveideaofafunction
Ú Constants,Variables
Ú Expressionsinvolving relationships
between two variables
Ú Functionsof asinglevariable
Ú Functionalnotation
Ú Domain, codomain and range
Ú One -one functions
Ú Ontofunctions
Ú Inversefunctions
Explains the evolution of the number
systems
Representsarealnumbergeometrically
Classifiesdecimalnumbers
Rationalisesthedenominator
ofexpressionswithsurds
Explainstheintuitiveideaofafunction
Recognizesconstants,variables
Relationshipbetweentwovariables
Explainsinversefunctions
ExplainDomain,Codomain
Explains One-onefunctions explains
ontofunctions
For Comments

9
No. of
Periods
2.2 Reviews types of functions • Types of functions
Ú Constantfunctions,linearfunctions,
piece-wisefunctions, modulus
(absolutevalue)function
• Graph of a function
• Compositefunctions
02
3. Analyses
quadratic
functions
3.1 Explores theproperties of
quadraticfunctions
3.2 Interprets the roots of a
quadraticequation
• Quadraticfunctions
Definitionofaquadraticfunction
2
( ) ; , ,
f x ax bx c a b c
    ;
and 0
a 
Ú Completingthesquare
Ú Discriminent
• Properties ofaquadraticfunction
Ú Greatest value, least value
Ú Existence/non-existenceofreal
zeros
Ú Graphsofquadraticfunctions
• Roots of a quadratic equation
Ú Sum and product of the roots
Ú Equations whose roots are
symmetricexpressionsof theroots
of aquadratic equation
Ú Natureof roots usingdiscriminant
Ú Condition fortwo quadratic
equations to havea common root
Ú Transformation of quadratic
equations
10
15
Recognizesspecialfunctions
Sketches the graph of a
functions
Findscompositefunctions
Introducesquadraticfunctions
Explainswhataquadraticfunctionis
Sketches the properties of a quadratic
function
Sketches the graph of a quadratic func-
tion
Describes the different types of graphs
ofthequadraticfunction
Describes zerosof quadradicfunctions
ExplaneingtheRootsofaquadratic
equation
Finds the roots of a quardatic equation
Expresses the sum and product of the
roots of quadraticequation in terms of
itscoefficient
Describes the nature of the roots of a
quardaticequation
Findsquadraticequationswhoserootsare
symmetricexpressionsof  and 
For Comments

10
No. of
Periods
01
01
05
4. Manipulates
Polynomial
functions
• Polynomialsofsinglevariable
polynomials
Ú Terms, coefficients, degree,
leadingterm,leadingcoefficient
• Addition,subtraction,multiplication,
divisionand longdivision
• Divisionalgorithm
• Syntheticdivision
• Remaindertheorem
• Factor theorem and its converse
• Solutionofpolynomialequations
4.1 Explorespolynomialsofa
singlevariable
4.2 Appliesalgebraicoperations
topolynomials
4.3 Solvesproblems using
Remainder theorem, Factor
theorem and its converse
Solves problems involving quadratic
functionsandquadraticequations
Transforms roots to otherforms
Definesapolynomialofasinglevariable
Distinguishesamonglinear,quadratic
andcubicfunctions
States theconditions for two
polynomialstobeidentical
ExplainsthebasicMathematical
operationsonpolynomials
Divides apolynomialbyanother
polynomial
States thealgorithmfordivision
States and prove remainder theorem
States Factor theorem
Expresses the converse of the Factor
theorem
SolvesproblemsinvolvingRemainder
theorem and Factor theorem.
Defineszerosofapolynomial
Solvespolynomialequations
( Order 4
 )
For Comments

11
No. of
Periods
5. Resolves
rational
functions
into partial
fractions
6. Manipulates
index
and
logarithmic
laws
5.1 Resolvesrationalfunction
into partial fractions
6.1 Uses index laws and
logarithmiclawstosolve
problems
• Rationalfunctions
Ú Proper andimproper rational
functions
• Partialfractionsof rationalfunctions
Ú Withdistinctlinearfactorsinthe
denominator
Ú Withrecurringlinearfactorsinthe
denominator
Ú With quadraticfactors in the
denominator (Up to 4 unknowns)
• Theindex laws
• Logarithmiclawsof base
• Change of base
06
01
Definesrationalfunctions
Defines properrational functions and
improperrationalfunctions
Finds partial fractionsof proper
rationalfunctions(upto4unknown)
Partialfractionsofimpropper
irationalfunction(upto4unknowns)
Uses index laws
Useslogarithmiclaws
Uses changeofbaseto solveproblems
For Comments

12
No. of
Periods
04
04
06
7. Solves
inequalities
involving
real
numbers
7.1 States basicproperties of
inequalities
7.2 Analysesinequalities
7.3 Solvesinequalitiesinvolving
modulus (absolutevalue)
function
• Basicpropertiesof inequalities
includingtrichotomylaw
• Numericalinequalities
Ú Representinginequalitiesonthereal
number line
Ú Introducting intervalsusing
inequalities
• Inequalitiesinvolving simplealgebraic
functions
Ú Manipulationoflinear,quadraticand
rational inequalities
Ú Finding the solutions of the above
inequalities
Ú algebraically
Ú graphically
• Inequalitiesinvolving modulli(absolute
value)
Ú Manipulationofsimpleinequalities
involvingmodulus(absolutevalue)
sign
Ú Solutionsof theaboveinequalities
Ú algebraically
Ú graphically
Definesinequalities
States thetrichotomylaw
Represents inequalitiesonareal
numberline
Denotesinequalities intermsof
intervalnotation
States andproves fundamental
resultsoninequalities
Solvesinequalitiesinvolvingalgebric
expressions
Solves inequalitiesincludingrational
functions,algebraicallyandgraphically
States the modulus (absolute value)
of areal number
Sketchesthegraphsinvolving
modulusfunctions
Solvesinequalitiesinvolvingmodulus
(onlyforlinearfunctions)
For Comments

13
No. of
Periods
8. Uses
relations
involving
angular
measures
9. Interpretes
trignometric
funtions
Ú Angulurmeasure
Ú Theangleanditssignconvention
Ú Degree and radian measures
Ú Length of acircular arc, S r

Ú Areaof acircularsector,
2
1
2
A r 

• Basictrigonometricfunctions
Ú Definitions of the six basic trigono-
metricfunctions,domain andrange
• Valuesofthecircularfunctionsof
theangles
0, , ,
6 4 3
  
and
2

8.1 Statestherelationship
between radians and
degres
8.2 Solves problems involving
arc length and area of a
circularsector
9.1 Describes basic
trigonometric(circular)
functions
9.2 Derives values of basic
trigonometricfunctionsat
commonlyusedangles
01
01
04
01
Introduces degrees and radians as
unitsofmeasurementofangles
Convert degrees into radian and
vice-versa
Find the lenth of an arc and area of a
circular sector
Explainstrigonometricratios
Defines basic trigonometric circular
functions
Introduces thedomains and the
rangesofcircularfunctions
Findsthevaluesoftrigonometric
functionsatgivenangles
States the sign of basic
trigonometric functionineach
quadrant
For Comments

14
No. of
Periods
9.3 Derives the values of basic
trigonometric functions at
angles differing byodd
multiples of
2

and integer
multiplesof 
9.4 Describes thebehaviourof
basictrigonometric
functionsgraphically
9.5 Findsgeneralsolutions
• Trignometricrelationsof theangle
3
, , , , 2
2 2
    
 
      
etc
• Graphs ofthebasictrigonometric
functions andtheirperiodicproperties
• Generalsolutionsoftheform
sin sin , cos cos and
tan tan
   
 
 

03
04
04
Describes theperiodic properties of
circularfunctions
Describes thetrigonometricrelations
of
(- ), 3
θ, θ, θ, 2 θ
2
2

  
   
in terms of θ
Finds thevaluesofcircularfunctions
atgivenangles
Represents the circular functions
graphically
Draws graphs of combined circular
functions
Solvestrigonometricequations
For Comments

15
No. of
Periods
10.Manipulates
trigonometric
identities
• Pythagoreanidentities
Ú Trigonometricidentities
• Sumanddifferenceformulae
Ú Applicationsinvolvingsumand
differenceformulae
• Product-sum,sum-product formulae
Ú Applicationsinvolving product-sum,
and sum - product formulae
• Doubleangle, tripleangle and half
angleformulae
• solutions ofequationsoftheform
cos sin , where , ,
a b c a b c
 
  
10.1 UsesPythagoreanidentities
10.2 Solvestrigonometric
problemsusing
sumanddifferenceformulae
problemsusingproduct-sum
and sum-productformulae
problemsusing
Doubleangles,Tripleangles
andHalfangles
04
02
05
03
Explainsanidentity
Explainsthedifferencebetweeniden-
titiesandequations
ObtainsPythagorianIdentities
SolvesproblemsinvolvingPythagorian
Identities
Constructsadditionformulae
Usesadditionformulae
Manipulates product - sum, and
Sum -product formulae
Solvesproblemsinvolving
sum - product,product-sumformulae
Solvesproblemsusingdouble, tripple
andhalfangles
Derives trigonometric formula for
double, tribleandhalfangles
Solves equations of the form
cos sin
a b c
 
  onlyfinding
solutions is expected)
For Comments

16
No. of
Periods
11.Applies
sine rule
and
cosine rule
to solve
trigonomet-
ricproblems
12. Solves
problems
involving
inverse
trigonomet-
ric
functions
11.1 States and proves sine rule
and cosinerule
11.2 Applies sineruleand
cosinerule
12.1 Describes inverse
trignometricfunctions
12.2 Represents inverse
functionsgraphically
12.3 Solvesproblemsinvolving
inversetrignometric
functions
• Sineruleand cosinerule
• Problemsinvolvingsineruleandcosine
rule
• Inversetrignometricfunctions
• Principalvalues
• Sketchinggraphsofinversetrignometric
functions sin-1
, cos-1
, tan-1
• Problemsinvolvinginverse
trigonometricfunctions
01
06
02
02
04
Introducesusual notationsforatriangle
States and prove sine rule for any
triangle
States and prove cosine rule for any
triangle
Solvesproblemsinvolvingsineruleand
cosinerule
Definesinversetrignometricfunctions
States the domain and the range of
inversetrigonometricfunctions
Draws the graph of an inverse
Solvessimpleproblemsinvolvinginverse
For Comments

17
No. of
Periods
• Intuitiveideaof
lim ( ) , where ,
x a
f x l a l

 
• Basictheoremsonlimitsandtheir
applications
• Proof of
1
lim
n n
n
x a
x a
na
x a


 


 

 
,
where n is a rational number and its
applications
• Sandwichtheorem(withoutProof)
• Proof of
0
sin
lim 1
x
x
x

 

 
 
and its
applications
13. Determines
thelimitof a
function
13.1 Explains thelimitofa
function
13.2 Solves problems using the
theoremsonlimits
13.3 Usesthelimit
1
lim
n n
n
x a
x a
na
x a


 


 

 
to solveproblems
13.4 Usesthelimit
0
sin
lim 1
x
x
x

 

 
 
to
solveproblems
02
03
03
03
Explainsthemeaningoflimit
Distinguishesthecaseswherethelimit
of afunction does not exist
. Expressesthetheorems on limits.
Proves
1
lim
n n
n
x a
x a
na
x a





where
n is arational number.
Solves problems involving above
result
States thesandwich theorem
Provesthat
0
sin
lim 1
x
x
x


Solvestheproblemsusingtheabove
result
For Comments

18
No. of
Periods
13.5 Interprets onesided limits
13.6 Findlimitsatinfinityandits
applicationstofindlimitof
rationalfunctions
13.7 Interpretsinfinitelimits
13.8 Interprets continuity at a
point
• Intuitive ideaofonesidedlimit
• Right hand limitandlefthand limit:
lim ( ), lim ( )
x a x a
f x f x
 
 
• Limit ofarationalfunctionas x  
Ú Horizontal asymptotes
• Infinitelimits
Ú Verticalasymptotesusingone
sidedlimits
• Intuitiveideaofcontinuity
02
02
01
02
Interprets onesided limits
Findsonesidedlimits ofagiven
functionatagivenrealnumber
Interpretslimitsat infinity
Explainshorizontalasymptotes
Explainsverticalasymptotes
Explains continuityat a point byusing
examples
For Comments

19
No. of
Periods
14.1 Describes the idea of
derivativeofa function
14.2 Determinesthederivatives
fromthefirstprinciples
14.3 States and uses the
theoremsondifferentiation
14.4 Differentiatesinverse
14. Differenti-
ates
functions
using suit-
able meth-
ods
• Derivativeas theslopeoftangentline
• Derivativeasalimit
• Derivative as arate of change
• Derivativesfromthefirstprinciples
Ú
n
x , where n is a rational number
Ú Basic trigonometricfunctions
Ú Functionsformedbyelementary
algebraic operations of the above
• Theoremsondifferentiation
Ú Constantmultiplerule
Ú Sumrule
Ú Product rule
Ú Quotientrule
Ú Chainrule
• Derivativesof inversetrigonometric
functions
Explains slopeand tangent at apoint
Definesthederivativeasalimit
Explainsrateofchange
• Findsderivativesfromthefirst
principles
States basicrules of derivative
Solves problems usingbasicrules of
derivatives
Findsthederivativesofinverse
trignometricfunctions
Solvesproblemsusingthederivatives
ofinversetricnomatricfunctions
06
05
03
03
For Comments

20
No. of
Periods
14.5 Describes natural exponen-
tialfunction andfind itsde-
rivative
• Theproperties of natural exponential
function
Ú  
x x
d
e e
dx

Ú Graph of
x
e
Definestheexponentialfunction(ex
)
Express domain and range
of exponentialfunction
States that eis an irrational number
Describes the properties of the ex
Writes theestimates ofthe valueof e
Writesthederivativeoftheexponential
function and usesit to solveproblems
The graph of y = ex
02
14.6 Describesnaturallogarithmic
function
• Propertiesofnaturallogarithmic
function
Ú Definitionof naturallogarithmic
function, ln x or  
log 0
e x x  ,
astheinversefunctionof
x
e , its domain and range
Ú  
1
ln , for 0
d
x x
dx x
 
Ú Graph of ln x
• Definition of x
a and itsderivative
Defines the natural logarithmic function
Expresses the domain and range
of the logarithmic function
Expresses the properties of lnx
The graph of y = lnx
Defines the function ax
for a > 0
Expresses the domain and the range
of y = a x
Solves problems involving logarithmic
function
Deduces the derivative of lnx
Deduces the derivative of
x
a
Solves problems using the derivatives of
lnx and
x
a
03
For Comments

21
No. of
Periods
14.7 Differentiatesimplicit
functionsandparametric
functions
14.8 Obtainsderivativesof
higher order
• Intuitiveideaof implicitfunctions
andparametricfunctions
• Diferentiation involvingImplicit
functionsandparametricequation
includingparametricformsatparabola
y2
= 4 ax and clipse
2 2
2 2
1
x y
a b
 
and hyperabola
• Successivedifferentiation
• Derivatives of higherorder
Definesimplictfunctions
Findsthederivativesofimplicitfunctions
Differentiatesparametricfunction
Writes down theequation ofthe
tangent andnormalatagivenpointtoa
givencurve
Finds derivativesofhigherorder
Differentiatesfunctionsofvarioustypes
Find relationshipamongvarious orders
ofderivatives
06
02
For Comments

22
No. of
Periods
• Stationarypoints
• Increasing/decreasingfunctions
• Maximumpoints(local),minimum
points(local)
• Pointtoinflection
• First derivativetest and second
derivativetest
• Concavityand points of inflection
• Sketchingcurvesonly(including
horizontalandverticalasymptotes)
• Optimizationproblems
15.1Investigatestheturning
pointsusingthederivative
15.2Investigatestheconcavity
15.3Sketches curves
15.4Appliesderivativesfor
practical situations
15. Analyses the
behaviour of
a function
using
derivatives
Definesstationarypointsofagivin
function
Describeslocal(relative)maximum
andalocalminimum
Employsthefirstderivativetesttofind
themaximumandminimumpointsofa
function
States that thereexists stationary
points which areneitheralocal
maximumnor alocalminimum
Introducespointsofinflection
Uses the second order derivative to
test whethera turning points of a
givenfunctionisalocalmaximumora
localminimum
Uses secondderivativeto find
concavity
Sketches the graph of a function
Uses derivativesto solve
reallifeproblems
05
02
04
04
For Comments

23
No. of
Periods
16.1DeducesindefiniteIntegral
usinganti-derivatives
16.2Uses theorems on integra-
tion
16.3Reviewthebasic properties
ofa definiteintegralusing
thefundamentaltheoremof
calculus
16.4 Integratesrationalfunctions
usingappropriatemethods
16.5 Integratestrigonometric
expressions using
trigonometricidentities
• Integration as the reverse process of
differentiation (anti-derivatives of a
function)
• Theoremsofintegration
• Fundamental Theorem of Calculus
• Intuitiveideaof thedefiniteintegral
• Definiteintegralanditsproperties
• Evaluation of definiteintegrals
• Indefiniteintegralsoffunctions of the
form
 
( )
f x
f x

; where  

f x is the
derivative of f (x) with respect to x
• Useofpartial fractions
• Useof trigonometricidentities
0000000000
Findsindefiniteintegralsusingthe
resultsofderivative
Uses theoremsonintegration
Usesthefundamental theoremof
calculus to solveproblems
Solvesdefiniteintegralproblems
Uses theproperties of definte
integral
Usestheformula
Uses ofpartial fractions for
integration
Uses trigonometricidentities forin-
tegration
u
03
02
02
05
03
16. Find
indefinite
and definite
Integrales
of functions
For Comments

24
No. of
Periods
• Integrationbysubstitution
• Integration byparts
• Usesofintegration
Ú Area under a curve
Ú Area between two curves
• Useoftheformulae  
 
2
b
a
f x dx


tofindthevolumeofrevolution
16.6 Uses themethod of
substitutionfor integration
16.7 Solveproblemsusing
integrationbyparts
16.8 Determines the area of a
region bounded by curves
usingintegration
16.9 Determines the volume of
revolution
04
03
04
02
Usessuitablesubstitutionstofind
intergrals
Uses integration byparts to solve
problems
Uses definiteintegralsto find area
under a curve and area between two
curves
Uses integration formula to find the
volumeofrevolution
For Comments

25
No. of
Periods
17. Uses the
rectangular
system of
Cartesian
axes and
geometrical
results
17.1 Finds thedistance between
two points on the Cartesian
plane
17.2 Finds Co-ordinates of the
pointdividingthestraight
linesegmentjoiningtwo
givenpointsinagivenratio
01
02
• RectangularCartesian coordinates
Ú RectangularCartesiansystem
Ú Distance betweentwo points
Coordinates ofthepoint that divides
alinesegmentjoiningtwogiven
pointsinagivenratio
Ú internally
Ú externally
Explains theCartesian coordinate
system
Defines the abscissaand the ordinate
Introducesthefourquadrantsinthe
cartesiancoordinateplane
Findsthelengthofalinesegment
joiningtwopoints
• Finds Co-ordinates ofthe point divid-
ingthestraightline segmentjoiningtwo
givenpointsinternallyinagivenratio
• Finds Co-ordinates ofthe point divid-
ingthestraightlinesegmentjoiningtwo
givenpointsexternallyinagivenratio
For Comments

26
No. of
Periods
• Straightline
Ú Inclination(angle),gradient(slope)
Ú Intercepts on the x and y axes
Ú Various forms ofequation ofa
straightline
• Point ofintersectionoftwo non
parallel straightines
• Theequation ofthestraight linepass-
ingthroughthepointofintersectionof
twogivennonparallelstraightlines
• Theconditionthatthetwogivenpoints
are on the same or oppositesides of a
givenstraightline
• Anglebetweentwostraightlines
• Therelationshipbetweenthegradients
of pairs of
Ú parallellinesÚ perpendicularlines
• Parametricequationofastraight line
• Perpendiculardistancefromapointto
astraightline
• Equations of bisectors of the angles
between twointersectingstraightlines
18. Interprets
the straight
line in terms
of Cartesian
co-ordinates
18.1 Derives theequation of a
straightline
18.2 Derives theequation of a
straightlinepassingthrough
thepointofintersectionof
twogivennonparallel
straightlines
18.3 Describetherelativeposition
oftwo pointswithrespect to
agivenstraightline
18.4 Finds theanglebetween
twostraightlines
18.5 Derives theperpendicular
distancefromagivenpoint
toagivenstraightline
05
02
02
02
06
Interpretsthegradient(slope)ofaline
and the x and y intercepts
Derivesvariousformsofequationofa
straightline
Findsthecoordinatesofthepoint of
intersectionof twononparallel
straightlines
Finds theequationofthelinepassing
throughthe
intersectionoftwogiven lines
Finds the Condition fortwo points to
beonthesamesideoranoppositsides
ofagivenline
Finds condition fortwo lines to be
parallelorperpendicular
Finds the angles between two given
linesbyusingtheirgradients
Derives parametric equation of a
straightline
Find perpendicular distance from a
point to agiven line using parametric
equationoftheline
Finds theequations ofangular
bisectors of two non parall straight
lines
For Comments

27
No. of
Periods
• Methodof mathematicalinduction
Ú PrincipleofMathematicalInduction
Ú Applicationsinvolving,divisibility,
summationandInequalities
• Sigmanotation
 
1 1 1
1 1
;where is a constant
  
 
   
 
  
 
n n n
r r r r
r r r
n n
r r
r r
U V U V
kU k U k
19. Applies the
principle
of
Mathemati-
calInduction
as a type of
proof for
Mathemati-
cal results
for positive
integers
20. Finds sums
of finite
series
19.1 Uses theprincipleof
Mathematical Induction
20.1 Describes finite series and
theirproperties
05
03
Statestheprinciplesof Mathematical
Induction
Provesthevariousresultsusing
principleof MathematicalInduction
Describesfinitesum
Uses the properties of “  ”
notation
For Comments

28
No. of
Periods
20.2 Finds sumsof elementary
series
21.1 Sumsseriesusingvarious
methods
Arithmetricseriesandgeometricseries
n
2 3
1 r=1 1
, r ,
 
  
n n
r r
r r andtheirapplications
• Summationofseries
Ú Method ofdifferences
Ú Methodofpartial fractions
Ú Principleof MathematicalInduction
05
08
03
Finds general term and the sum ofAP,
GP,
Proves and uses theformulae for
values of
n
2 3
1 r=1 1
, r ,
 
  
n n
r r
r r to find
thesummationofseries
Uses various methodsto find the
sum ofa series
Interprets sequences
Finds partialsumof aninfiniteseries
Explains theconcepts of convergence
anddiverganceusingpartialsums
Finds the sum ofa convergent series
21 Investigates
infinite series
21.2 Usespartialsumtodetemine
convergenceand divergence
• Sequences
• Partialsums
• Concept of convergence and
divergence
• Sumtoinfinity
• Sequences
For Comments

29
No. of
Periods
03
06
02
22.1 Describes the basic
properties ofthe binomial
expansion
22.2 Appliesbinomialtheorem
23.1 Uses the Complex number
system
• Binomialtheoremforpositiveintegral
indices
Ú Binomialcoefficients, generalterm
Ú Proof of the theorem using
mathematicalInduction
• Relationshipsamongthebinomial
coefficients
• Specificterms
• Imaginaryunit
• Introduction of ,thesetofcomplex
numbers
• Real part and imaginarypart of a
complexnumber
• Purely imaginarynumbers
• Equalityof twocomplex numbers
23. Interprets
the system
of complex
numbers
Statesbinomialtheoremforpositive
integralindices.
Writesgeneraltermandbinomial
coefficient
Proves thetheorem using
MathematialInduction
Writestherelationshipamongthe
binomialcoefficients
Findsthespecificterms of binomial
expansion
Statestheimaginaryunit
Definesacomplex number
States the real part and imaginary
part at a complex number
Uses theequalityoftwo complex
numbers
22. Explores
thebinomial
expansion
for positive
integral
indices
For Comments

30
No. of
Periods
•Algebraicoperationsoncomplexnumbers
1
1 2 1 2 1 2
2
, , ,
z
z z z z z z
z
  
2
( 0)
z 
• Definitionof z
• Proofs ofthefollowingresults:
Ú
1 2 1 2
  
z z z z
Ú
1 2 1 2
  
z z z z
Ú
1 2 1 2
  
z z z z
Ú
1 1
2 2
 
 
  
 
   
z z
z z
• Definition of z ,modulus ofa
complex numberz
• Provesofthefollowingresults:
Ú
1 2 1 2
  
z z z z
Ú
1
1
2
2 2
if 0
 
z
z
z
z z
Ú
2
 
z z z
Ú
2 2 2
1 2 1 1 2 2
2Re( )
z z z z z z
    
applications oftheabove results
23.2 introducesalgebraic
operations on complex
numbers
23.3 Proves basic properties
ofcomplex conjugate
23.4 Definethemodulusofa
complexnumber
02
02
04
Defines algebraicoperations on
complexnumbers
Uses algebraic operations between
two complex numbers and verifies
that theyare also complex numbers
Basic operations or algebraic opera-
tions
Defines z
Obtains basicproperties ofcomplex
conjugate
Proves the properties
Defines the modulus 0



of a
complexnumberz
Proves basic properties of
modulus
Applies thebasicproperties
For Comments

31
No. of
Periods
• TheArganddiagram
• Representing z x iy
  by the point
( , )
x y
• Geometricalrepresentationsof
1 2 1 2
, , ,
z z z z z z

 
where  
• Polar form ofa non zero complex
number
• Definitionofarg( z )
• DefinitingArg z , principalvalueofthe
argument z is thevalueof  satisfying
  
  
• Geometrical representationof
Ú
1
1 2 2
2
, ; 0
z
z z z
z
 
Ú (cos sin )
 

r i , where
  , r > 0
Ú
1 2
z z
 
 


, where ,
  
and 0
 
 
23.5 Ilustratesalgebraic
operationsgeometrically
usingtheArganddiagram
04
Represents the complex number on
Arganddiagram
Contstructs points representing
z1
+z2
, z and  z where  
Expresses anonzerocomplex number
in polaForm
(cos sin ): 0,
z r i r
  
   
Definestheargumentofacomplex
number
Definestheprincipleargumentofanon
zerocomplex number
Constructspoints representing
z1
z2
and
1
2
z
z in theArgand diagram
(cos sin )
r i
 
 where
, 0
R r
  
1
z zc
 
 


, where , R
   and
0
 
 
For Comments

32
No. of
Periods
• proofof thetriangleinequality
1 2 1 2
z z z z
  
• Deductionof reversetriangleinequality
1 2 1 2
z z z z
  
• State and prove of the DeMovier’s
Theorem
• ElementaryapplicationsofDeMovier’s
theorem
• Locus of
Ú 0
z z k
  and 0
z z k
 
Ú  
0
Arg  
z z  a n d
 
0
Arg z z 
 
where   
   and 0
z is fixed
Ú 1 2
z z z z
   , where 1 2
and
z z
aregivendistinct complex numbers
02
04
23.6 Uses the DeMovier’s
theorem
23.7 Identifies locus / region of a
variablecomplexnumber
Provesthetriangleinequality
Deduces thereverse triangle
inequality
Uses the above inequalities to solve
problems
States and prove of the DeMovier’s
Theorem
Solvesproblemsinvolvings
Elementaryapplicationsof
DeMovier’s theorem
Sketchs the locus of variable com-
plex numbersinArganddiagram
Obtains the Cartesian equation of a
locus
For Comments

33
No. of
Periods
01
02
06
24.1 Definesfactorial
24.2 Explainsfundamental
principlesofcounting
24.3 Useofpermutationsasa
techniqueofsolving
mathematicalproblems
• Definition of n ! , the factorial n for
n 
 or n = 0.
Ú Generalform
Ú Recursiverelation
• Techniquesregardingtheprinciplesof
counting
• Permutations
Ú Definition
Ú Thenotation
n
r
P andtheformulae
When 0 ;
r n Z
 
  
24.Uses
permutations
and
combinations
as
mathematical
models
for sorting
and
arranging
Definesfactorial
States therecursiverelation for
factorials
Explainsthefundamentalprincipleof
counting
Defines
n
r
P and obtain the
formulae for
n
r
P .
Thenumberofpermutationsofndif-
ferent objects taken r at a time
Findsthenumberpermutationsof
different objects taken all time at a
time
Permutation of n objectsnot all
different
Explainsthecyclicpermutations
Findsnumerofpermutationsof n dif-
ferent objects not all different taken
r at a time
For Comments

34
No. of
Periods
24.4 Uses combinations as a
techniqueofsolvingmath-
ematicalproblems
• Combinations
Ú Definition
Ú Define as
n
r
C and finds a formulae
for
n
r
C
Ú Distinctionbetweenpermutationand
combnation
Definescombination
Define as
n
r
C and finds a formulae for
n
r
C
Finds the number of combinations of
n different objects taken r at a time
where  
0
r r n
 
Explainsthedistinctionbetween
permutationsandcombinations
Definesamatrix
Definestheequalityofmatrices
Definesthemultiplicationofamatrixby
a scalar
Explainsspecialtypesofmatrices
Uses theaddition ofmatrices
Writestheconditionforcompatibility
25.1 Describesbasicpropertiesof
matrices
25. Manipulates
matrices
055
02
• Definitionandnotation
Ú Elements,rows,columns
Ú Sizeof amatrix
Ú Rowmatrix,columnmatrix,
squarematrix,nullmatrix
• Equalityof two matrices
• Meaning of A
 where  is a scalar
Ú Properties of scaler product
Ú Definitionofaddition
Ú Properties ofaddition
For Comments

35
No. of
Periods
• Subtractionsofmatrices
• Multiplicationofmatrices
Ú Compatibility
Ú Definitionofmultiplication
Ú Propertiesofmultiplication
• Squarematrices
Ú Order of a square matrix
Ú Identity matrix, diagonal matrix,
symmetric
matrix,skewsymmetricmatrix
Ú Triangularmatries (upper, lower)
02
03
04
25.2 Explains special cases of
squarematrices
Identifies the orderof a square
matrices
Classifies the different types of
matrices
25.3 Describes the transpose
and theinverseofamatrix
25.4 Uses matrices to solve
simultaneousequations
• Transposeofa matrix
Ú Definitionandnotation
• Inverse of a matrix
Ú Onlyfor 2 2
 matrices
• Solution of apair oflinearequations
with two variables
Ú Solutionsgraphically
Ú The existenceof aunique
solutions,infinitely many solutions
andnosolutionsgraphically
Finds the transposeof a matrix
Finds theinverseof a2x2 matrix
Solvessimultaneousequations
usingmatrices
Illustratesthesolutionsgraphically
Definessubtractionusingadditionand
scalermultiplication
Writestheconditionsforcompatibility
States and uses the properties of
multiplicationtosolveproblems
For Comments

36
No. of
Periods
• Equation ofacirclewithoriginas
the cetre and a given radius
• Equation of acirclewithagiven
centre and radius
• Conditions that acircle and a
straight lineintersects,touches ordo
notintersect
• Equation ofthe tangent to acircle at
a pointon circle
• Equationtangentdrawntoacirclefrom
anexternalpoint
• length oftangent drawn from an
extenal point to a circle
• Equation ofchord of contact
03
02
03
26.1 Finds theCartesian
equation of a circle
27.1 Describes theposition of a
straightlinerelativetoa
circle
27.2 Finds the equations of tan-
gents drawn to acircle from
anexternalpoint.
26. Interprets
the
Cartesian
equation of
a circle
27.Explores
Geometric
properties
of circles
Defines circle as a locus of a
variablepointsuchthatthedistance
from afixedpoint is aconstant
Obtains the equation ofa circle
Interprets thegeneral equation of
acircle
Finds theequationofthecircle
havingtwo givenpoints as the
end points at a diameter
Discusesthepositionofastraight
line with respect to acircle
Obtains the equationof thetangent
at a point on a circle
Obtains theequation ofthe
tangent drawn to a circle from an
externalpoint
Obtains the length of tangent
drawn from an external point to a
circle
Obtains theequationofthechord
of contact
For Comments

37
No. of
Periods
27.3 Derivesthegeneral
equation of a circle
passingthroughpointof
intersection of a given
straightlineandagivencircle
27.4 Describes the position of
two circles
27.5 Finds the condition for two
circle to intersect
orthogonally
• Theequationofacirclepassingthrough
thepointsofintersection
of a straight line and a circle
• Position of two circles
Ú Intersection of two circles
Ú Non-intersection of two circles
Ú Twocirclestouchingexternally
Ú Twocirclestouchinginternally
Ú Onecirclelyingwithintheother
• Conditionfortwocirclesto enterseet
orthogonaly
Interprets the equation
S + U = 0
Discribes theconditionfortwo
circles to Intersect or Not-intersect
Discribes theconditionfortwo
circles to TouchexternallyorTouch
internally
DiscribesTo haveonecirclelying
withinthe othercircle

Finds the conditionfortwo circles to
intersectorthogonally
20202
02
03
02
For Comments

38
No. of
Periods
1.1 Investigates vectors
03
• Introduction of scalar quantities and
scalars
• Introduction of vector quantities and
vectors
• Magnitudeand direction of avector
• Vectornotation
Ú Algebraic,Geometric
Ú Nullvector
• Notationformagnitude(modulus)ofa
vector
• Equalityoftwo vectors
• Trianglelawofvectoraddition
• Multiplyingavectorbyascalar
• Definingthe differenceoftwo
vectors as a sum
• Unitvectors
• Parallelvectors
Ú Condition for two vectors to be
parallel
• Addition of three or more vectors
• Resolution ofavector in any
directions
COMBINED MATHEMATICS - II
Explainsthediffernecesbetween
scalarquantitiesand scalars
Explainsthediffernecebetween
vector quantityand a vectors.
Represents avectorgeometrically
Expresses the algebraicnotation of a
vector
Defines themodulusofavector
Definesthenullvector
Defines -a , where a is a vector
States the conditions for two vectors
to be equal
States thetrianglelawofaddition
Deduces the parallelogramlawof
addition
Adds three or more vectors
Multiplies a vectorbya scalar
Subtracts a vector from another
Identifies theanglebetween two
vectors
Identifiesparallelvectors
States the conditions for two vectors
to be paralles
1. Manipulates
Vectors
For Comments

39
No. of
Periods
1.2 Constructs algebraic
system forvectors
1.3 Applies positionvectors to
solve problems
• laws for vector addition and
multiplicationbyascaler
• Positionvectors
• Introduction of i and j
 
• Position vector relative to the 2D
Cartesian Co-ordinate system
• Applicationofthefollowingresults
Ú If a and b are non - zero and
non - parallel vectors and if
0
a b
 
 

then
0 and 0
 
 
State the conditions for two vectors
to beparallel
Definesa“unitvector”
Resolves a vectorin a given
directions
States the properties of addition and
multiplicationbyascaler
Definespositionvectors
Expressesthepositionvectorofapoint
intermsofthecartesianco-ordinatesof
thatpoint
Addsandsubtractsvectorsintheform
xi y j

Provesthat if a, b aretwononzero,
non - parallel vectors and if
0
a b
 
  then 0
  and
0
 
Applications of theabove results
01
06
For Comments

40
No. of
Periods
04
02
Defines the scalar product of two
vectors
States that the scalar product of two
vectors is a scalar
States the properties of scalar
product
Interprets scalar product
geometrically
Solvessimplegeometricproblems
involvingscalarproduct
Define vectorproduct of two
vectors
States the properties of vector
product
(Application of vector product are
not expected)
Describes the concept of a particle
Describes the concept of a force
Statesthataforceisalocalizedvector
Represents aforcegeometrically
Introduces different types of forces
inmechanics
Describes the resultant of a system
of coplaner forces acting at a point
• Definition ofscalarproduct oftwo
vectors
• Properties of scalar product
Ú a b b a
   (Commutativelaw)
Ú  
a b c a b a c
      (Distributive
law)
• Conditionfortwo non-zerovectorsto
be perpendicular
• Introduction of k
• Definition ofvectorproduct oftwo
vectors
• Properties of vector product
Ú    
a b b a
• Concept of a particle
• Concept of a force and its
representation
• Dimension andunit of force
• Types of forces
• Resultant force
1.4 Interprets scalar and vector
product
2.1 Explains forces acting on a
particle
2. Uses
systems of
coplanar
forces
For Comments

41
No. of
Periods
2.2 Explains the action of two
forces actingon a particle
2.3 Explains the action of a
systems of coplanar forces
actingonaparticle.
• Resultant oftwo forces
• Parallelogramlaw of forces
• Equilibriumundertwoforces
• Resolution of a force
Ú intwogivendirections
Ú in twodirectionsperpendicularto
each other
• Define coplanarforcesactingon a
particle
• Resolvingthesystemofcoplanarforces
intwodirectionsperpendiculartoeach
other
• Resultant of the system of coplanar
forces
Ú method of resolution of forces
Ú graphicalmethod
04
04
Statestheparall-elogramlawofforces
tofindtheresultantof twoforces
actingatapoint
Uses the parallelogram law to obtain
formulaetodeterminetheresultantof
two forces acting at a point
Solvesproblemsusingthe
parallelogram lawofforces
Writes the condition necessaryfor a
particletobeinequilibriumundertwo
forces
Resolves a given force into two
componentsintwogivendirections
Resolves a given force into two
components perpendicular to each
other
Determines the resultant of three or
morecoplanarforcesactingat apoint
byresolution
Determinesgraphicallytheresultantof
three or more coplanar forces acting
at a particle
States the conditions for a system of
coplanarforces actingon aparticleto
beinequilibrium
For Comments

42
No. of
Periods
Writestheconditionforeqilibrium
(i) 0
0
0, 0
R
R Xi Yj
X Y

  
 
Completes apolygon of forces.
Explainswhatismeantbyequilibrium.
Statestheconditionsforequilibriumof
a particle under the action of three
forces
Statesthetheoremoftriangleofforces,
forequilibriumofthreecoplanarforces
States the converse of the theorem of
triangleofforces
StatesLami’stheoremforequilibrium
of three coplanar forces acting at a
point
Proves Lami’sTheorem.
Solvesproblemsinvolvingequilibrium
of three coplanar forces acting on a
particle
2.4 Explainsequilibriumofa
particleundertheaction
of three forces.
• Conditionsforequilibrium
Ú null resultantvector
0
R X i Y j
  

Ú Vector sum 0
 or, equivalently,
,
0 and 0
 
X Y
Ú Completion of Polygonof forces
• Triangle Law
• Lami’sTheorem
• ProblemsinvolvingLami’stheorem
05
For Comments

43
No. of
Periods
04
2.5 ExplainstheResultant of
coplanar forces acting on
a rigid body
• Concept of a rigid body
Ú Principleof transmissionof forces
Ú Explainingthetranslationaland
rotational effectofa force
Ú Forces acting on a rigid body
Ú Defining the moment of a force
about a point
Ú Dimensionandunitofmoment
Ú Physicalmeaningofmoment
Ú Magnitudeandsenseofmomentof
a force about a point
Ú Geometricinterpretationofmoment
• General principle about moment of
forces
Ú Algebraicsumof themomentsofthe
component forcesabout apoint on
the plane of a system of coplanar
forces is equvalent to moment of
the resultant forceaboutthatpoint
Describes arigid body
Statestheprincipleof transmissionof
forces
Explains thetranslation and rotation
of a force
Defines the moment of a force about
apoint
Explainsthephysicalmeaningof
moment
Finds the magnitude of the moment
about a point and its sense
States the dimensions and units of
moments
Represents themagnitudeof the
moment of a force about a point
geometrically
Determines the algebraic sum of the
moments of the forces about a point
in the plane of a coplanar system of
forces
Usesthegeneralprincipleofmoment
of asystem of forces
Usestheresultantoftwoparallelforces
acting on a rigitd body
For Comments

44
No. of
Periods
2.6 Explainstheeffectoftwo
parallel coplanar forces
acting on a rigid body
• Resultant of two forces
Ú When thetwoforces arenotparallel
Ú Whenthetwoforcesareparalleland
like
Ú When two forces of unequal
magnitudeareparallelandunlike
• Equilibriumunder twoforces
• Introduction of a couple
• Moment of a couple
Ú Magnitudeandsense of themoment
of a couple
Ú The moment of a couple is
independentofthepoint aboutwhich
themomentis taken
• Equivalence of two coplanar couples
• Equilibriumundertwocouples
• Composition ofcoplanarcouples
Uses the resultant of two non -
parallelforcesactingonarigidbody
States theconditions forthe
equilibriumoftwoforcesactingona
rigidbody
Describes a couple
Calculates themomentofacouple
States that the moment of a couple
is independent of the point about
which the moment of the forces is
taken
coplanar couples to beequivalent
coplanar couples to balance each
other
Combines coplanarcouples
06
For Comments

45
No. of
Periods
08
2.7 Analyses asystem of
coplanar forces
Reduces a couple and a single force
actinginitsplaneintoasingleforce
Shows that a force acting at a point is
equivalent to the combination of an
equal forceactingat another point
togetherwithacouple
Reduces a system of coplanar forces
to asingle force acting at an arbitrary
point Oand a couple ofmoment G
Reduces any coplanar system of
forces to a single force and a couple
actingat anypoint in that plane
(i) Reduces of a system of coplanar
forces to a single force
 
X 0 or Y 0
 
(ii) Reduces of a system of forces to
a couple when X= 0. Y = 0 and
G 0

(iii) Expresses conditions for
equilibrium
Findsthemagnitude,directionandthe
line ofaction of theresultant ofa
coplanar system of forces
• A force (F) acting at a point is
equivalent to a force F acting at any
given pointtogetherwith acouple
• Reducing a system of coplanar forces
toasingleforceRactingatagivenpoint
together with acoupleof moment G
• Magnitude,directionand lineofaction
ofthe resultant
• Conditions forthereduction of system
of coplanar forces to
Ú asingleforce:
 
R 0 X 0 or Y 0
  
Ú a couple :
 
R = 0 X=0 and Y 0

and G 0

Ú equilibrium
R=0 (X=0 and Y=0) and G=0
Ú Single force at other point : 0

R ,
G 0

• Problemsinvolvingequlibriumofrigid
bodies under the action of coplanar
forces
For Comments

46
No. of
Periods
08
10
10
2.8 ExplainstheEquillibriumof
three coplanar forces
2.9 Investigates the effect of
friction
2.10 Applies the properties of
systems of coplanar forces
toinvestigateequilibrium
involvingsmoothjoints
Statesconditionsfortheequilibriumof
threecoplanarforcesactingonarigid
body
Finds unknown forces when a rigid
bodyisinequillibrium
• Allforcesmust beeitherconcurrentor
all parallel
• Use of
Ú Triangle Law of forces and its
converse
Ú Lami’stheorem
Ú Cotangentrule
Ú Geometrical properties
Ú Resolvingintwoperpendicular
directions
• Introduction ofsmooth andrough
surfaces
• Frictional forceand its nature
• Advantages anddisadvantages of
friction
• Limitingfrictionalforce
• Lawsoffriction
• Coefficientoffriction
• Angleoffriction
• Problemsinvolvingfriction
• Types ofsimplejoints
• Distinguishamovablejointandarigid
joint
• Forces acting at a smooth joint
• Applicationsinvolving jointedrods
Describes smoothsurfaces and rough
surfaces
Describesthenatureoffrictional force
Explains theadvantages and
disadvantagesoffriction
Writesthedefinitionof limiting
frictionalforce
States thelaws offriction
defines theangleoffrictionandthe
coefficientoffriction.
Solvesproblemsinvolving friction
States thetypeof simple joints
Describes themovablejoints and rigid
joints
Marks the forces acting on a smooth
joints
Solvestheproblemsinvolving joined
rods
For Comments

47
No. of
Periods
08
04
2.11 Determines the stresses in
therodsofaframeworkwith
smoothlyjointed rods
• Frameworks with light rods
• Conditionsfortheequilibriumateach
joint at the framework
Ú Bow’s notation and stress
diagram
Ú Calculationofstresses
2.12 Appliesvarious
techniques to determine
the centreof mass of
symmetrical
uniformbodies
• Definitionofcentreofmass
• Centre of mass of a plane body
symmetricalaboutaline
Ú Uniformthinrod
Ú Uniformrectangularlamina
Ú Uniformcircularring
Ú Uniformcirculardisc
• Centre of mass of a bodysymmetrical
about a plane
Ú Uniformhollowor solidcylinder
Ú Uniformhollowor solid sphere
• Use of thin rectangularstripesto find
the centreof mass of aplane lamina
and useofit in findingthecentreof
massof thefollowinglamina
Ú Uniformtriangularlamina
Ú Uniformlaminaintheshapeofa
parallelogram
. Describesaframeworkwith lightrods
States theconditionfortheequilibrum
at each joint in the frame work
Uses Bow’s notation
Solvesprobleminvolingaframework
withlightrod
Defines thecentre of mass ofa system
of particlesin aplane
Defines thecentre ofmass ofalamina
Finds thecentreofmass ofuniform
bodies symmetricalabout aline
Finds the centre of mass of bodies
symmetrical aboutaplane
Finds centreof mass of aLammina of
differentshapes
For Comments

48
No. of
Periods
2.13 Finds the centre of mass of
simplegeometricalbodies
usingintegration
2.14 Finds thecentre of mass
(centreofgravity)of
composite bodies and
remainingbodies
2.15 Explainscentreofgravity
2.16 Determinesthestabilityof
bodiesinequilibrium
2.17 Determinestheangleof
inclinationofsuspended
bodies
• Centreofmassofuniformcontinuous
symmetricbodies
Ú Circular arc, circular sector
• Thecentre ofmass of uniform
symmetricbodies
Ú Hollow rightcircularcone
Ú Solidrightcircularcone
Ú Hollowhemisphere
Ú Solidhemisphere
Ú Segment of a hollowsphere
Ú Segment of a solid sphere
• Centre of mass of composite bodies
• Centre ofmass of remaining bodies
• Introduction ofcentreofgravity
• Coincidence ofthecentre ofgravity
and centre of mass
• Stabilityofequilibriumofbodiesresting
on a plane
• problemsinvolvingsuspendedbodies
Finds thecentreofmass ofsymmetrical
bodiesusingintegration
Finds thecentreof massofcomposite
bodies
Findsthecentreofmass ofremaining
bodies
States the centre of mass and centre of
gravityaresameundergravitational
field.
Explainsthestabillityofbodiesin
equilibriumusingcentreofgravity
Solvesprobleminvolvingsuspended
bodies
06
04
02
02

49
No. of
Periods
3. Applyes the
Newtonian
model to
describe the
instanta-
neous
motion in a
plane
3.1 Uses graphs to solve
problemsinvolvingmotion
inastraightline
• Distanceandspeedandtheirdimensions
andunits
• Average speed, instantaneous speed,
uniformspeed
• Position coordinates
• Displacementandvelocityandtheir
dimensionsandunits
• Averagevelocity,instantaneous
velocity,uniformvelocity
• Displacement -timegraphs
Ú Average velocitybetween two
positions
Ú Instantaneous velocityat apoint
• Averageacceleration,itsdimensionsand
units
• Instantaneousacceleration,
uniform accelerationandretardation
• Velocity-timegraphs
• Gradient of the velocity time graph is
equal to the instantaneous acceleration
atthatinstant
Defines“distance”
Defines average speed
Defines instantaneousspeed
Definesuniformspeed
Statesdimensionsandstandardunitsof
speed
States that distance and speed are
scalarquantities
Defines positioncoordinates ofa
particleundergoingrectilinearmotion
DefinesDisplacement
Expresses thedimension and standard
unitsofdisplacement
Definesaveragevelocity
Definesinstantaneous velocity
Definesuniformvelocity
Expresses dimension andunitsof
velocity
Draws thedisplacement
timegraphS
For Comments

50
No. of
Periods
• Theareasignedbetweenthetimeaxis
and the velocitytime graph is equal
to thedisplacement described during
thattimeinterval
08
Finds theaverage velocitybetween
twopositionsusingthedisplacement
timegraph
Determinestheinstantaneousvelocity
usingthedisplacementtimegraph
Definesaccelaration
Expressesthedimensionand unitof
acceleration
Defines averageacceleration
Definesinstantaneousacceleration
Definesuniformacceleration
Defines retardation
Draws thevelocitytime graph
Findsaverageaccelaration usingthe
velocitytimegraph
Finds theacceleration at a given
instantusingvelocity-timegraph
Findsdisplacementusingvelocitytime
graph
Draws velocitytimegraphsfor
differenttypesofmotion
Solvesproblemsusingdisplacement
timeandvelocity-timegraphs
For Comments

51
No. of
Periods
Derives kinematic equations for a
particlemovingwithuniform
acceleration
Derives kinematic equations using
velocity-timegraphs
Useskinematicequationsforvertical
motionundergravity
Uses kinematics equations to solve
problems
Usesvelocity-timeanddisplacement
- time graphs to solve problems
Describes the concept of frame of
referencefortwodimensionalmotion
Describes the motion of one body
relativeto anotherwhen two
bodiesaremovinginastraightline
States theprincipleofrelative
displacementfortwobodiesmoving
alongastraightline
velocityfortwobodiesmovingalong
astraightline
• Derivation of constant acceleration
formulae
Ú Usingdefinitions
Ú Usingvelocity-timegraphs
1 2 2 2
, , , 2
2 2

      
 
 
 
t
u v
v u at s s ut at v u as
• Verticalmotionunderconstant
acceleration due to gravity
Ú Use ofgraphs and kinematic
equations
• Frame of reference for one
dimensionalmotion
• Relativemotioninastraightline
• Principle of relative displacement,
relative velocity and relative accel-
eration
• Useofkinematicequationsandgraphs
whenrelativeaccelerationis
constant
3.2 Uses kinematicequations to
solveproblemsinvolving
motion in astraight linewith
constant acceleration
3.3 Investigates relative motion
between bodies moving in a
a straight line with constant
accelerations
08
06
For Comments

52
No. of
Periods
3.4 Explainsthemotionofa
particle on a plane.
3.5 Determinestherelativemo-
tion of two particles mov-
ing onaplane
• Position vectorrelativetotheorigin
ofamovingparticle
• Velocityandaccelerationwhen the
position vector is given as a function
oftime
• Frame of reference
• Displacement,velocityand
acceleration relative toa frame of
reference
• Introducerelativemotionoftwo
particlesmovingon aplane
• Principlesofrelativedisplacement,
relative velocity,andrelative
acceleration.
• Path of aparticle relative to another
particle
• Velocityofaparticlerelativeto
anotherparticle
acceleration for two bodies moving
alongastraightline
Uses kinematicequations and graphs
related to motion for two bodies
movingalongthesamestraightlinewith
constantrelativeacceleration
Finds relation between thecartesian
coordinates and the polar coordinates
of apointmovingon aplane
Finds the velocity and acceleratain
when the position vector is givin as a
functionoftime
Definestheframeofreferance
obtains thedisplacement and velocity
and acceleration relative to frame of
referance
Explains the principles of relative
displacement,relativevelocity,and
relativeacceleration
Finds thepath and velocityrelative to
anotherparticle
06
06
For Comments

53
No. of
Periods
3.6 Usesprinciplesofrelative
motion to solve real word
problems
3.7 Explainsthemotionofa
projectileinavertical plane
• Shortest distancebetween two
particles and the time taken to reach
the shortest distance
• Thetimetakenand position when
two bodies collide
• Time taken to describe a given path
• Use of vectors
• Given theinitialpositionandthe
initial velocityof aprojected particle
thehorizontalandvertical
components of
(i) velocity (ii) displacement,aftera
timet
• Equation of thepath of a projectile
• Maximumheight
• Timeofflight
• Horizontalrange
Ú Two angles of projection which
givethesamehorizontalrange
Ú Maximum Horizontal range
Uses the principles of relative motion
to solves theproblems
Findstheshortestdistancebetweentwo
particles
Finds therequirementsforcollision of
two bodies
Introduces projectile
Describes theterms “velocityof
projection”and“angleofprojection”
States that the motion of a projectile
can be considered as two motions,
separately, inthehorizontaland
verticaldirections
Applies thekinematicequationsto
interpret motionofaprojectile
Culculatesthecomponentsofvelocity
of a projectile afteragiven time
Findsthecomponentsofdisplacement
of a projectilein agiven time
10
08
For Comments

54
No. of
Periods
Calculatesthemaximumheightof a
projectile
Culculates he time taken to reach the
maximumheightofaprojectile
Calculates thehorizontalrangeofa
projectileanditsmaximum
States that in general there are two
angles of projection for the same
horizontal range fora given velocityof
projection
Findsthemaximumhorizontalrangefor
a given speed
Foragivenspeedofprojectionfindsthe
angleofprojectiongivingthemaximum
horizontalrange
Derives Cartesian equationsofthepath
of a projectile
Findsthetimeofflight
Finds the angles of projection to pass
throughagivenpoint
For Comments

55
No. of
Periods
States Newton’s firstlawof motion
Defines“force”
Defines“mass”
Defineslinearmomentumofaparticle
Statesthatlinearmomentumisavector
quantity
Statesthedimensionsandunitoflinear
momentum
Describesaninertialframeofreference
States Newton’s secondlawofmotion
Defines Newton astheabsoluteunit of
force
Derives the equation F = ma from
second lawofmotion
Explains thevectornatureof theequa-
tion F = ma
Statesthegravitationalunits offorce
Explains the difference between mass
and weight of a body
Describes “action”and “reaction”
States Newton’s third lawofmotion
Solvesproblemsusing F ma

Bodies incontact and particles
connectedbylightinevtenciblestrings
Systemofpullys,wedges(maximum4
pullys)
3.8 Applies Newton’s laws to
explainmotionrelativetoan
inertialframe
• Newton’s firstlawofmotion
• concept ofmass linear
momentum and inertialframeof
reference
• Newton’s second law of motion
• Absoluteunitsandgravitionalunitsof
force
• Distinguishbetweenweightandmass
• Newton’s thirdlawofmotion
• Application ofnewton’s laws(under
constantfoceonly)
• Bodies in contact and particales
connectedbylightinextensiblestring 15
For Comments

56
No. of
Periods
08
Explains the concept of work
Defines work doneunder a constant
force
States dimension andunits ofwork
ExplainsEnergy
Explainsthemechanicalenergy
DefinesKineticEnergy
DefinesPotentialEnergy
Explains the Gravitational Potential
Energy
ExplainstheElasticPotentialEnergy
Explainsconservativeforces and
dissipativeforce
Writes work - energyequations
Explains conservationofmechanical
Energyandappliestosolveproblems
States dimension andunitsofenergy
3.9 Interpretsmechanicalenergy • Definitionofwork
work done byconstant force
Dimensionandunitsofwork
• Introduceenergy,itsdimensions and
units
• Kineticenergyasa typeofmechanical
energy
Definition of kinetic energyof a
particle
work energyequation for kinetic
energy
• Disssipativeandconservativeforces
• Potential energyas atypeof mechanical
energy
Definition of potential energy
Definitionofgravitationalpotential
energy
work energyequation forpotential
energy
• Definitionofelasticpotentialenergy
• Expressionfortheelasticpotential
energy
• The work done bya conservative force
is independent ofthe path described
• Principleofconservationofmechanical
energyandits applications
For Comments

57
No. of
Periods
08
05
10
04
3.10Solves problems involving
power
• Definitionofpoweritsdimensionsand
units
• Tractive force( F)constant case only
• Definitionandapplicationof
Power= tractiveforcex velocity
( P=F.V)De
• Impulse as a vector its dimension and
units
• I=  (mv) Formula
• Loss of kineticenergydue to an
impulsiveactione
DefinesPower
Statesitsunits anddimensions
Explainsthetractiveforce
Derives theformulaforpower
Uses tractive force when impluse is
constant
ExplainstheImpulsiveaction
Statestheunitsand Dimensionof
Impulse
Uses I =  mv to solve problems
FindsthechangeinKineticenergydue
toimpulse
• Newton’s lawofrestitution
• Coefficientofrestitution  , 0 1
e e
 
• Perfect elasticity  
1
e 
• Loss of energywhen 1
e 
• Direct impact of two smooth elastic
spheres
• Impact of asmooth elastic sphere
moving perpendicularto aplane
Explainsdirectimpact
States Newton's law of restitution
Definescoefficientofrestitution
Explains the direct impact of a sphere
on afixed plane
Calculateschangeinkineticenergy
Solvesproblemsinvolvingdirect
impacts
3.11Interpretstheeffectofanim-
pulsive action
3.12Uses Newton’s law of resti-
tutionto directelasticimpact
Solvesproblemousingthepriciple
oflinearmomentum
• Principleofconservationoflinear
momentum
3.13Solvesproblemsusingthe
conservationoflinear
momentum
For Comments

58
No. of
Periods
06
04
10
3.14Investigatesvelocityand
accelerationformotion
incirculer
3.15Investigatesmotionina
horizontalcircle
• Angularvelocity  andangular
acceleration 
 of a particale moving
on a circle
• Velocityand acceleration ofa particle
moving on acircle
• Motion of a particle attached to an end
ofalight inextensiblestringwhose
other end is fixed, on a smooth
horizontalplane
• Conicalpendulum
Defines the angular velocity and
accelaration ofaparticalemovingin a
circle
Find the velocityand the acceleration
ofaparticlemovinginacircle
• Finds the magnitude and direction of
theforceon aparticle movingin a
horizontalcirclewithuniformspeed
• solves theproblems involvingmotion
inahorizontalcircle
• solvestheproblemsinvolving
conicalpendulam.
• Explainsverticalmotion
• Explains the motionofa ringthreeded
onafixedsmooth vertical wire/
particle moving in a fixed smooth
circular,verticaltube
• Finds thecondition forthemotionof a
particlesuspendedfromaninelasticlight
stringattached toafixed point,in
verticalcircle.
• Discussesthemotionofaparticleonthe
outersurfaceofafixedsmoothspherein
averticalplane
• Solvesproblemsincludingcircular
motion.
• Applications of lawofconservation of
energy
• uses the law F ma

• Motion ofaparticle
Ú on the surface of a smooth sphere
Ú insidethehollow smooth sphere
Ú suspendefromaninextensible,light
stringattached to afixed point
Ú threaded in afixed smooth circular
vertical wire
Ú In avertical tube
3.16 Investigatestherelevent
principles formotion ona
verticalcircle
For Comments

59
No. of
Periods
04
06
06
3.17 Analyses simple harmonic
motion
3.18 Describes the nature of a
simpleharmonicmotionona
horizontalline
3.19 Describes the nature of a
simple harmonicmotion on
averticalline
• Definitionofsimpleharmonicmotion
• Charactersticequationofsimple
harmonicmotion,anditssolutions
• Velocityasafunctionofdisplacement
• Theamplitudeand period
• Displacementasafunctionoftime
• Interpretationofsimpleharmonic
motionbyuniformcircularmotion,
andfindingtime
• UsingHooke’s law
Ú Tensioninaelasticstring
Ú Tension orthrustin aspring
• Simple harmonic motion of a particle
under theaction of elasticforces
• Simpleharmonicmotionofaparticle
on avertical line undertheaction of
elasticforces andits own weight
• Combinationof simpleharmonic
motionandfreemotionundergravity
Defines simple harmonic
mortion(SHM)
Obtain thedifferential equation of
simpleharmonicmotionand
verifiesitsgeneralsolutions
Obtains the velocityas a function
ofdisplacement
Defines amplitude and period of
SHM
Describes SHM associated with
uniform circularmotion and finds
time
Findsthetensionin anelasticstring.
TensionorthustinaspringusingHookes
Law
DescribesthenatureofsimpleHarmonic
motiononahorizontalline
ExplainsthesimpleHarmonicmotion
onaverticalline
Solvesproblemwithcombination
ofsimpleharmonicmotionandmotion
undergravity.
For Comments

60
No. of
Periods
04
06
4. Applies
mathemati-
cal
models to
analyse
random
events
4.1 Interprets events of a random
experiment
4.2 Applies probabilitymodels to
solve problems on random
events
• Intuitiveideaofprobability
• Definitionofarandomexperiment
• Definition of samplespace and sample
points
Ú Finitesamplespace
Ú Infinitesamplespace
• Events
Ú Definition
Ú Simpleevent,compositeevents,
nullevent
complementaryevents,
Ú Union of two events, intersection of
two events
Ú Mutuallyexclusiveevents
Ú Exhaustiveevents
Ú Equallyprobableevents
Ú Event space
• Classicaldefinitionof probabilityandits
limitations
• Frequencyapproximationtoprobability,
anditslimitations
• Axiomaticdefinitionof probability, and
itsimportance
Explainsrandomexperiment
Defines samplespace and sample
point
Definesanevent
Explains eventspace
Explainssimpleeventsand
compound events
Classifiestheeventsfindsunionand
intersectionofevents
Statesclassicaldefinitionofprobabil-
ityanditslimitations
Statestheaxiomaticdefinition
Proves the theorems on probability
usingaxiomaticdefinitionandsolves
problems usingtheabove theorems.
For Comments

61
No. of
Periods
08
• Theoremsonprobabilitywith proofs
Ú Let A and B be any two events in a
given sample space
(i)    
1
P A P A
   where A is
the complementryevent of A
(ii) Addition rule
Ú        
P A B P A P B P A B
    
Ú If ,
A B
 then    
P A P B

• Definitonofconditionalprobability
• Theorems with proofs let 1 2
, , ,
A B B B
be anyfour events is agiven sample
space with   0
P A  . then
(i)   0
 
P A
(ii)    
1 ,
P B A P B A
  
(iii)      
/
1 1 2 1 2
   
P B A P B B A P B B A
(iv)      
1 2 1 2
     
 
P B B A P B A P B A
 
1 2 )

P B B A
• Multiplicationrule
• If 1 2
,
A A areanytwo events in a given
sample spacewith  
1 0
P A 
then      
1 2 1 2 1
  
P A A P A P A A
4.3 Applies theconcept of
conditionalprobabilityto
determinetheprobabilityofa
random eventunder given
conditions
Definesconditionalprobability
States and proves the theorems on
conditionalprobability
statesmultiplicationrule
States
Frequencyappoximationto
probabilityanditslimitation.
Importanceofaxiomaticdetinition.
For Comments

62
No. of
Periods
04
06
4.4 Uses theprobabilitymodel to
determinetheindependenceof
two or three events
4.5 Applies Bayes theorem
• Independence of two events
• Independence ofthree events
• PairwiseIndependence
• MutuallyIndependence
• Partition of a sample space
• Theoremontotalprobability,withproof
• Baye’sTheorem
Uses independent two or three
events to not solveproblems
Defines apartition of a sample space
States and prove on theorem of total
probability
States Baye’s theorem and applies to
leave problems
For Comments

63
No. of
Periods
01
03
04
5. Applies sci-
entific tools
to develop
decision
making skills
5.1 Introduces thenatureofstatis-
tics
5.2 Describes measuresofcentral
tendency
5.3 Interpretsafrequencydistribu-
tionusingmeasuresofrelative
positions
• Definitionof statistics
• Descriptivestatistics
• Arithmetricmean,modeandmedian
• Ungrouped data
• Datawithfrequencydistributions
• Groupeddatawithfrequencydistri-
butions
• Weightedarithmetricmean
• Median, Quartiles and Percentiles for
ungrouped and grouped data with fre-
quencydistributions
BOX Plots
Explainswhatisstatistics
Explainsthenatureofstatistics
Finds thecentral tendency
measurments
Describes the mean,median and
mode as measures of central
tendency
Findstherelativepositionof
frequencydistribution
For Comments

64
No. of
Periods
08
02
5.4 Describes measureof disper-
sion
5.5 Determines theshape of a
distributionbyusingmeasures
of skewness.
• Introduction to Measures of disper
sionandtheirimportancy
• Types of dispersion measurements
Ú Range
Ú InterQuartilerangeand Semi
inter-quartilerange
Ú meandeviation
Ú varianceandstandard deviation
Ú ungrouped data
Ú ungrouped datawith frequency
distributions
Ú group datawith frequency
distrubutions
• pooled mean
• pooled variance
• z - score
• Introduction to Measures of skewness
• Karl Pearson’s measures of skewness
Usessuitablemeasuredispersionto
make decisions on frequencydis-
tribution
States the measures of dispersion
andtheirimportancy
Explain pooled mean, variance at
Z-score
Defines themeasureof skewness
Determines the shapes of the distribu-
tion usingmeasures ofskewness
For Comments

65
For Comments
8.0 TEACHING LEARNING STRATEGIES
To facilitatethe students to achievethe anticipated outcome ofthis course, a varietyof teachingstategies must beemployed. If studentsare to improve
theirmathematicalcommunication,forexample,theymusthavetheopportunitytodiscussinterpretations,solution,explanationsetc.withotherstudents
aswell astheirteacher.Theyshouldbeencouraged to communicatenotonlyinwritingbutorally,andto usediagrams aswell asnumerial,symbolicand
wordstatementsintheirexplanations.
Students learnin amultitudeofways. Students can bemainlyvisual, auditoryorkinesthetic learners, oremployavarietyofsenses when learning. The
range of learningstyles in influenced bymanyfactors, each ofwhich needs to be considered in determining the most appropriate teaching strategies.
Research suggests thatthe cltural and socialbackground has a significant impact on the waystudents learn mathematics. Thesedifferences need to be
recognised andavarietyofteachingstrategiesto beemployedso that all studentshaveequal accesstothedevelopment ofmathematicalknowledgeand
skills.
Learningcan occurwithin a largegroupwhere the class istaught as a wholeandalso within asmall gruop wherestudents interactwith othermembers
ofthegroup, orat anindividual level whereastudent interacts with theteacheroranotherstudent,or works independently.Allarrangements havetheir
placeinthemathematicsclassroom.

66
For Comments
9.0 SCHOOL POLICY AND PROGRAMMES
To make learning of Mathematics meaningful and relevant to the students classroom work ought not to be based purely on the development of
knowledgeandskillsbutalsoshouldencompass areas likecommunication,connection,reasoningand problemsolving.Thelatterfouraims,ensurethe
enhancement ofthethinkingand behaviouralprocessofchildern.
Forthispurposeapart fromnormalclassroom teachingthefollowingco-curricularactivitieswillprovidetheopportunityforparticipationofeverychild
inthelearningprocess.
Ú Student’s studycircles
Ú MathematicalSocieties
Ú Mathematicalcamps
Ú Contests(nationalandinternational)
Ú Useofthelibrary
Ú TheclassroomwallBulletin
Ú Mathematicallaboratory
Ú Activityroom
Ú Collectinhistoricaldataregardingmathematics
Ú Useofmultimedia
Ú Projects
Itistheresponsibilityofthemathematicsteachertoorganisetheaboveactivitiesaccordingtothefacilitiesavailable.Whenorganisingtheseactivitiesthe
teacher and thestudents can obtain theassistanceofrelevant outsidepersons and institution.
In orderto organise such activites on a regular basis it is essential that each school develops apolicyof its own in respect of Mathematics. This would
form a part of the overall school policy to be developed by each school. In developin the policy, in respect of Mathematics, the school should take
cognisanceofthephysical environmentoftheschool andneighbourhood,theneeds andconcernsofthestudentsandthecommunityassociatedwith the
school and theservices of resource personnel and institutions to which theschool has access.

MATHEMATICAL SYMBOLS AND NOTATIONS
The following Mathematical notation will be used.
1. Set Notations
 anelement
 notanelement
1 2
{ , , . . . }
x x the setwith elements 1 2
, , . . .
x x
 
/ ...
x or  
:...
x the set of all x such that...
(A)
n thenumberofelements in setA
 emptyset
 universalset
A/ the complement ofthesetA
thesetofnatural numbers,
{1, 2, 3, ...}
the set of integers  
0, ±1, ±2, ±3, . . .
 the set of positive integers {1, 2, 3, . . . }
thesetofrational numbers
the set of real numbers
thesetofcomplex numbers
 a subset
 a proper subset
M not subset
 not a proper subset
 union
 intersection
]
,
[ b
a thecolsed interval { : }
x R a x b
  
]
,
( b
a theinterval }
:
{ b
x
a
R
x 


)
,
[ b
a theinterval }
:
{ b
x
a
R
x 


)
,
( b
a theopen interval }
:
{ b
x
a
R
x 


2. Miscellaneous Symbols
 equal
 notequal
 identicalorcongruent
approximatelyequal
 proportional
 lessthan
Ÿ less than orequal
> greaterthan
¦ greater than or equal
 infinity
 ifthen
 ifandonlyif(iff)
For Comments

3. Operations
b
a  a plus b
b
a  a minus b
,
a b a b
  a multiplliedby b
,
a
a b
b
 a divided by b
b
a : the ratio between a and b
1
n
i
i
a

 1 2 ... n
a a a
  
a the positivesquareroot of the positivereal number a
a the modulus ofthereal number a
!
n n factorial where  
0
n 
 
!
( )!
n
r
n
P
n r


0 r n
  n 
 , {0}
r 
 
!
,0
!( )!
n
r
n
C r n
r n r
  

n 
 , {0}
r 
 
4. Functions
)
(x
f thefunctinof x
f:AB f is a function where each element of setAhas an
uniqueImageinset B
y
x
f 
: thefunction fmaps theelement xto theelement y
1
f 
theinverseofthefunctionf
 
g f x
 the compositefunction of gof f
lim ( )
x a
f x

the limit of  
f x as xtends to a
x
 an incrementof x
dy
dx
the derivativeof y with respect to x
n
n
d y
dx
the
th
n derivative of ywith respect to x
     
 
(1) (2)
, , . . . , n
f x f x f x
the first, second , ..., nth
derivatives of  
f x
with respect to x
 ydx indefiniteintegral of y with respectto x

b
a
ydx definiteintegralofyw.r.t x intheinterval a x b
 
, , ...
x x
  the first, second,... derivative of x with respect to
time
For Comments

5. Exponential and Logarithmic Functions
x
e exponential function of x
a
log x logarithm of x to the base a
ln x naturallogarithmof x
lgx logarithm of xto base 10
6. Cricular Functions
sin, cos, tan
cosec, sec, cot



thecircularfunctions
1 1 1
1 1 1
sin , cos , tan
cosec , sec , cot
  
  





theinversecircularfunctions
7. Complex Numbers
i the square root of - 1
z acomplex number, z x i y
 
 
co s sin
r i
 
 
Re( )
z the real part of  
,
z Re x i y x
 
Im( )
z theimaginarypart of  
, Im
z x i y y
 
z themodulusof z
 
arg z Theargumentof z
 
Arg z theprincipleargumentof z
z thecomplex conjugateof z
8. Matrices
M amatrix M
T
M the transposeofthe matrix M
-1
M theinverseofthematrix M
det M thedeterminant ofthematrix M
9. Vectors
or
a a the vector a
AB


thevactorrepresented in magnitudeand direction bythe
directedlinesegmentAB
,
i ,
j k unit vectors in thepositivedirection ofthe cartesian axes
a the magnitudeof vectora
AB


themagnitude of vectorAB
a b the scalar product of vectors a and b
a × b the vetor product of vrctors aand b
For Comments

10. Probability and Statistics
A, B, C ect.. events
A B
 union of theeventsAand B
A B
 intersection oftheeventsAand B
P(A) probabilityoftheeventA
A/ complement oftheeventA
(A B)
P x probabilityof theeventAgiven that event Boccurs
X, Y, R, ... randomvariables
, , , ...
x y r ect. values ofthe random variables X,Y, R etc.
1 2
, , ...
x x observations
f1
, f2
, ... frequencieswithwhichtheobservations
1 2
, , ...
x x occur
x Mean
2
 Variance
 / S / SD Standarddeviation
For Comments

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